1 IR Spectroscopy

“Time is nature’s way of keeping everything

from happening at once.” - Woody Allen

The absorption of visible, infrared andfar infrared radiation can excite moleculesthrough electronic, vibrational and rotationaltransitions. In fact one can define the stateof a molecule in terms of its electronic, vi-brational and rotational states; a given elec-tronic state is subdivided into energy statescorresponding to the vibrational levels, eachof which is further subdivided into rotationallevels, with each taking progressively less en-ergy to excite. As a result their spectralsignatures end up at different spectroscopicregions: signatures due to electronic tran-sitions appear generally at UV and visiblewavelengths while vibrational features extendfrom optical to middle IR wavelengths andsignatures due to pure rotational transitionsend up in the far infraredThis is a very quick, somewhat qualitative,

review of nature of molecular spectroscopy.This review will give you enough material tocalculate the radiative transfer effects of gasesonce others have derived the line parameters,but not, of course, to derive line parametersyourselves.

2 Planetary Spectra

Let’s ignore, for now, the effects of scatter-ing by the gas (Raleigh) and particulates and

consider only the absorption due to gas.

The giant planets and Titan all have simi-lar optical and near-IR spectra, because theyare all dominated by methane (CH4) vibra-tional bands. Methane, therefore, plays alarge roll in establishing the pressure lev-els where most of the solar insolation is ab-sorbed. The visible to near-IR spectra ofVenus and Mars indicate CO2 features, asdoes that of Earth. But Earth distinguishesitself with water, oxygen, and in the UV,ozone signatures, which sets it apart fromother planets.

Consider the infrared spectra of planetaryatmospheres. The thermal emission of theouter planets and Titan is dominated in partby stratospheric emission features due largelyto photochemically produced species, createdin part from the dissociation of methane.Two primary coolants are ethane (C2H6) andacetylene (C2H2). Titan’s atmosphere is alsocooled by emission from hydrogen cyanide(HCN), which is produced from the dissoci-ation of its main constituent, N2. Anothercomponent of Titan’s infrared spectrum isthe pressure-induced absorption of N2, H2and CH4, which creates a greenhouse effectthat warms the surface by 21 K. In contrast,the spectra of the inner rocky planets exhibitabsorption by CO2 and, for Earth, a hostof other constituents such as water, nitrousoxide (N2O), and ozone. Earth is quite theweirdo. We’ll get back to that later.

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Figure 1: Molecular electronic, vibra-tion, and rotation transitions.

2.1 Rotation Spectra.

Molecular rotation can be envisioned classi-cally as the rotation of a semi-rigid moleculeabout it’s center of mass. In classical me-chanics, the rotational energy, ER, dependson angular momentum, L, as:

ER =1

2Iω2 =

L2

2I

where ω is the angular velocity and I the mo-ment of inertia1. Quantization of angular mo-mentum associates L with a quantized value

1For example: for a diatomic molecule of reduced

mass µ=m1m2/m1+m2, and internuclear separationR: I = µR2

of ~J , where J is an integer (the quantumnumber) and ~ is h/2π, such that:

ER(J) =~2

2IJ(J + 1). (1)

The energy difference between two succes-sive levels (~2/I) is generally on the orderof 10−4 to 10−3 eV. At room temperature,the translational thermal energy of molecules(3/2kBT ) is 2.5×10

−2 eV. Therefore colli-sions in planetary atmospheres transfer thenecessary energy of excitation, and broadrange of energy states are occupied. In fact,for temperatures typical of planetary atmo-spheres, the rotational state populations obeythe Boltzmann distribution and are thus inLTE. As we shall see later on, fluorescencemore readily occurs for vibrational and elec-tronic transitions, which require higher en-ergy collisions to excite.

Since rotational transitions occur fromthe interaction of the electric dipole of themolecule and the electric field of the incidentradiation, diatomic molecules of identical nu-clei, like N2 and O2, do not exhibit pure ro-tation spectra. These molecules lack electricdipole moments. That said these moleculescan be perturbed sufficiently to attain tempo-rary dipole moments that allow for collisioninduced spectra.

Note that the separation between spec-tral lines, ∆( 1

λ) = ~/2πcI, is constant with

wavenumber Thus the value of I and, fordiatomic molecules, the internuclear separa-tion, R, can be measured from the spacingbetween spectral lines.

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2.2 Vibration-Rotation Spec-tra.

The molecular moment of inertia, I, is nothowever a fixed entity. Instead it changeswith rotation with the extension of the in-ternuclear distance. In addition it changeswith the vibration of the molecule.According to classical mechanics, the vi-

bration of a semi-rigid system can be decom-posed into a set of normal modes. The num-ber of vibrations possible for a molecule gen-erally increase with the number of atoms, N ,it contains. As a rule of thumb, for moleculesof N>2, there are 3N -6 independent modesfor a non-linear molecule and 3N -5 for a lin-ear molecule.If the vibrations are small in amplitude as

occurs for the lowest energy states, the oscil-lations can be approximated as simple har-monic, for which the quantized states are:

EV = hνk(v + 1/2) (2)

where νk is the mode frequency and vk isan integer - the vibrational quantum num-ber. The mode is denoted by the vibrationalconstant, k. Another term is also used, thevibrational constant, ωek=νk/c. The separa-tion between states is constant for a simpleharmonic oscillator and equal to hνk. Thelowest energy state is not zero, because ofthe uncertainty principal. For each particu-lar mode the frequency spacing between thevibrational states is constant and equal to themode frequency νk if the simple harmonic ap-proximation applies.Consider CO. There is only only one vi-

bration mode, that of stretching along the

Figure 2: Vibration modes of some com-mon molecules. Note that the ν1 and ν2modes of CH4 are inactive.

plane of the molecule. The mode frequencyν1∼2127 cm

−1, where the 1→0 transitionlines occur. The 2→1 transitions occur atthe same wavelength (roughly) and the 2→0transitions occur at twice this frequency. Wecan estimate the frequencies of the lines ofhigher transitions, only these transitions de-viate further from the simple harmonic oscil-lator assumption.

In fact since the rotation transitions ac-company vibration transitions, the energy

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levels are written in terms of an anharmonicoscillator and the interaction between the ro-tation and vibration transitions, e.g.

E(v, J) = hc[ωe(v + 1/2)− ωexe(v + 1/2)2

+BvJ(J + 1)−DvJ2(J + 1)2],

the second term is a higher order vibrationalanharmonic term and the latter two termsare interaction terms (as indicated by the vsubscript and the J quantum number).The selection rule of v= ±1 applies gener-

ally. But there are also transitions where v=±2, ±3 and so on, a result of the anharmonicnature of most transitions.The rotational transitions provide a fine

structure to the spectrum of a particular vi-bration transition, with spectral lines havingwavenumbers smaller than the band head,the P-branch, and longer than the band head,the R-branch. These are formed from thetransition of the lower rotational states J ′′

to the higher state J ′ by:

∆J = J ′−J ′′ = +1 R−branch

∆J = J ′−J ′′ = −1 P−branch.

The R− branch transitions to a higher rota-tional energy level at the highest vibrationallevel, requiring higher energy level photonsthan P − branch.There is a further correction to the rota-

tional energies derived above. It turns outthat the electron angular momenta about theinternuclear axis of a diatomic molecule arecomparable to the nuclear value. Only thecomponent of the angular momentum, Λ, onthe spin axis remains constant. If Λ 6= 0 then

Figure 3: The R and P branches ofvibration-rotation transitions.

the rotational selection rules are ∆J = 0, ±1,and the Q branch is allowed. If Λ= 0, then∆J = ±1. Generally Q branches do not oc-cur in diatomic molecules. Q branches occurin spherical top molecules like methane.

3 Line Strengths

The strengths of spectroscopic lines dependson the probability of the individual transi-tion, but also on the population of the statesthat are involved. Here we’re going to as-sume LTE conditions. We therefore return

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to the Boltzmann distribution of states. Theratio of a particular rotational state’s popula-tion to the population of all rotational statescombined is:

n(J)

n=

(2J + 1)e−EJ/kBT

QR,

where:

QR =∑

J ′

(2J ′ + 1)e−(~2/2I)J ′(J ′+1)/kBT

is the partition function. The term 2J + 1is the degeneracy of the rotational state J ,a quantum mechanical result. If the spacingof the states is small compared to the extentof the rotational band, one can integrate thatabove expression in terms of dJ and one findsthe approximation that QR ∼ kBT/(~

2/2I).Note this ratio provides us with a probabil-ity that the particular lower energy rotationalstate will be populated. We use this ratio, in-stead of n1 when defining the line strength.The line strength is defined in terms of the

cross section σν of an individual vibration-rotation line:

S =

∫

σνφ(ν)dν.

The expression of the LTE cross section interms of the Einstein coefficients is:

σ̄ν =hν

4πB12

n̄1n(1− exp(−hν/kT )),

where we have used the Einstein coefficientsthat are defined in terms of the intensity.Let’s generalize to a multi-level atom anduse Bi rather than B12 and write in full the

Figure 4: The populations of rotationstates control relative intensities of ro-tational lines, which thus can serve asan atmospheric thermometer.

expression for the population of rotationalstates.

σ̄ν =hνBi4πQi

(2J + 1)e−EJ/kBT (1− e−hν/kT ).

Generally, data bases of line parametersgive you the line intensity at a particularreference temperature, T0, somewhere nearroom temperature. In order to determine theline intensity at a temperature of interest, T ,you must take into account the different pop-ulation of states at T0 and T , and the temper-ature dependence of the stimulated emissionterm. The line intensity at T can then beexpressed in terms of the line intensity at T0as:

σ̄ν(T ) = σ̄ν(T0)QV (T0)QR(T0)e

−E/kBT (1− e−hνi/kBT )

QV (T )QR(T )e−E/kBT0(1− e−hνi/kBT0)

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Figure 5: The shape, qualitativelyspeaking, of the R (right) and P (left)branches of a diatomic molecule’s vi-bration spectrum.

Here we see that the ratio of the line inten-sities depends on the ratio of the fractionalnumber of particles occupying the lower en-ergy state, and the ratio of the stimulatedemission, which acts as a negative absorp-tion. Note that both of these effects dependon temperature. The degeneracy of the statedoesn’t depend on temperature, so its ratiois unity.

The partition functions of the vibrationtransitions are usually close to one. Oftenthese values do not need to be considered.One exception to the rule is the ν9 vibrationalband of C2H6. Another exception is wa-ter. The values of QV R(T) = QV (T) QR(T)were derived by Vidler & Tennyson (J. Chem.Phys. 113, 9766, 2000).

As we saw above, the rotational partitionfunction, QR(T ), depends roughly on the firstpower of temperature. A better approxima-tion is the following:

QR(T ) ∼ T linear molecules

QR(T ) ∼ T3/2 non−linear molecules

The expression for the line intensity is thus:

σ̄ν(T ) = σ̄ν(T0)〈T0T〉m

e−E/kBT (1− e−hνi/kBT )

e−E/kBT0(1− e−hνi/kBT0),

where m depends, as shown above, on themolecular geometry. Note that this simpleexpression for the ratio of rotational partitionfunctions does not apply well for all moleculesand conditions, e.g. it is way off for water atthe temperatures (500-2000 K) characteristicof extrasolar “hot Jupiter” planets.

4 Practical Application

Molecular line bases give you the wavelengthof the intensity, the lowest energy state, E= E ′′, in units of cm−1, the line intensityσ̄ν(T0) = S in units of cm

−1/(molecule -cm−2), and the pressure broadening coeffi-cient. With this information you can calcu-late the absorption coefficient for vibration-rotation transitions. These units are definedin terms of wavenumber, νi, which is relatedto the frequency of the transition, f and thespeed of light, c, as νi = f/c. Also note thatsince E is the wavenumber units it must bemultiplied by hc to get the energy. Secondnote the units of intensity. The absorptioncoefficient depends on Si φ(νi) where φ(ν)is the line profile function, which is in unitsof 1/cm−1, that is inverse wavenumber. Thecombined value of Si φ(νi) is in units of in-verse column abundance, which are the finalunits of the absorption coefficient.

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5 Appendix

So where do equations 1 and 2 come from?Well there is not enough time to go into thewhole story, but this will give you an idea. In1926 Erwin Schrodinger formulated an equa-tion, call aptly the Schrodinger Equation thatis essentially an equation of motion, like New-ton’s equation, but for quantum mechanics -the physics of the microscopic states in na-ture. His equation can be written as:

−~2

2m

δ2Ψ(x, t)

δx2+ V (x)Ψ(x, t) = i~

δΨ(x, t)

δt

If the potential energy function is indepen-dent of time, we can simplify the equationusing a separation of variables. First we writeΨ(x, t) = ψ(x)φ(t). Then we plug this intothe equation and rearrange it to see if we canget all of the x dependent variables on oneside and the t dependent on the other. In-deed we can. Then the common variable,call it G, is not dependent on either x or tand we end up with two equations, whichare the 2 sides of the equation that are setequal G. From this exercise we get the time-independent Schrodinger Equation:

−~2

2m

δ2ψ(x)

δx2+ V (x)ψ(x) = Eψ(x),

andφ(t) = e−iEt/~.

Now consider a simple harmonic oscillator.The potential function for this system in aclassical world is:

V (x) =k

2x2,

which oscillates about an equilibrium posi-tion with a frequency ν of

ν =1

2π

√

k/m,

where m is the mass.In quantum mechanics, the solutions the

time-independent Schrodinger Equation forthis potential is:

En = (n+ 1/2)hν,

where ν is given above. So the the first eigen-value is:

E0 =1

2hν

The first eigenfunction is:

ψ0 = A0e−µ2/2,

where

µ =(km)1/4

~1/2x

You can check this by plugging it into thetime-independent Schrodinger Equation.

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Figure 6: A Sample of the HITRAN database.

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